C₁₅₁×#9005 #50313930

order	38656
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₅₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₅₁×#9005

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₅₁	#120
commutator	C₄²	C₂²×C₄×C₁₅₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₅₁
socle	C₂²×C₁₅₁	#386
prime radical	#532×C₁₅₁	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₅₁ ◁ C₂×C₄²×C₁₅₁ ◁ G

416 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 199
transitive degree	μ_t(G) = 19328

A minimal faithful permutation representation — 3 generators on 199 points:

(1,2,5,10,6,8,4,9)(3,11,14,7,13,16,12,15)(17,18)(19,28,31,39)(20,32)(21,26)(22,36)(23,27)(24,30,40,35)(25,33)(29,44,47,38)(34,41)(37,42)(43,45,48,46)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199);(1,3)(2,7)(4,14)(5,12)(6,13)(8,15)(9,16)(10,11)(17,19,21,35,23,31,22,30)(18,24,26,39,27,40,36,28)(20,29,33,46,34,47,37,45)(25,38,41,48,42,44,32,43);(2,8)(4,5)(7,15)(12,14)(17,20)(18,25)(19,29)(21,33)(22,37)(23,34)(24,38)(26,41)(27,42)(28,43)(30,45)(31,47)(32,36)(35,46)(39,48)(40,44)

Conjugacy classes & character table

6040 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38656, 1993);